(* *)
(**************************************************************************)
-include "Basic_2/grammar/term_vector.ma".
+include "basic_2/grammar/term_vector.ma".
(* GENERIC RELOCATION WITH PAIRS ********************************************)
inductive at: list2 nat nat → relation nat ≝
| at_nil: ∀i. at ⟠ i i
| at_lt : ∀des,d,e,i1,i2. i1 < d →
- at des i1 i2 → at ({d, e} :: des) i1 i2
+ at des i1 i2 → at ({d, e} @ des) i1 i2
| at_ge : ∀des,d,e,i1,i2. d ≤ i1 →
- at des (i1 + e) i2 → at ({d, e} :: des) i1 i2
+ at des (i1 + e) i2 → at ({d, e} @ des) i1 i2
.
interpretation "application (generic relocation with pairs)"
(* Basic inversion lemmas ***************************************************)
-fact at_inv_nil_aux: ∀des,i1,i2. @[i1] des ≡ i2 → des = ⟠ → i1 = i2.
+fact at_inv_nil_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 → des = ⟠ → i1 = i2.
#des #i1 #i2 * -des -i1 -i2
[ //
| #des #d #e #i1 #i2 #_ #_ #H destruct
]
qed.
-lemma at_inv_nil: ∀i1,i2. @[i1] ⟠ ≡ i2 → i1 = i2.
+lemma at_inv_nil: ∀i1,i2. @⦃i1, ⟠⦄ ≡ i2 → i1 = i2.
/2 width=3/ qed-.
-fact at_inv_cons_aux: ∀des,i1,i2. @[i1] des ≡ i2 →
- ∀d,e,des0. des = {d, e} :: des0 →
- i1 < d ∧ @[i1] des0 ≡ i2 ∨
- d ≤ i1 ∧ @[i1 + e] des0 ≡ i2.
+fact at_inv_cons_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 →
+ ∀d,e,des0. des = {d, e} @ des0 →
+ i1 < d ∧ @⦃i1, des0⦄ ≡ i2 ∨
+ d ≤ i1 ∧ @⦃i1 + e, des0⦄ ≡ i2.
#des #i1 #i2 * -des -i1 -i2
[ #i #d #e #des #H destruct
| #des1 #d1 #e1 #i1 #i2 #Hid1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1/
]
qed.
-lemma at_inv_cons: ∀des,d,e,i1,i2. @[i1] {d, e} :: des ≡ i2 →
- i1 < d ∧ @[i1] des ≡ i2 ∨
- d ≤ i1 ∧ @[i1 + e] des ≡ i2.
+lemma at_inv_cons: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 →
+ i1 < d ∧ @⦃i1, des⦄ ≡ i2 ∨
+ d ≤ i1 ∧ @⦃i1 + e, des⦄ ≡ i2.
/2 width=3/ qed-.
-lemma at_inv_cons_lt: ∀des,d,e,i1,i2. @[i1] {d, e} :: des ≡ i2 →
- i1 < d → @[i1] des ≡ i2.
+lemma at_inv_cons_lt: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 →
+ i1 < d → @⦃i1, des⦄ ≡ i2.
#des #d #e #i1 #e2 #H
elim (at_inv_cons … H) -H * // #Hdi1 #_ #Hi1d
lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd
elim (lt_refl_false … Hd)
qed-.
-lemma at_inv_cons_ge: ∀des,d,e,i1,i2. @[i1] {d, e} :: des ≡ i2 →
- d ≤ i1 → @[i1 + e] des ≡ i2.
+lemma at_inv_cons_ge: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 →
+ d ≤ i1 → @⦃i1 + e, des⦄ ≡ i2.
#des #d #e #i1 #e2 #H
elim (at_inv_cons … H) -H * // #Hi1d #_ #Hdi1
lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd